Cancellation for (G,n)(G,n)-complexes and the Swan finiteness obstruction

In previous work, we related homotopy types of $(G,n)$-complexes when $G$ has periodic cohomology to projective $\mathbb{Z} G$ modules representing the Swan finiteness obstruction. We use this to determine when $X \vee S^n \simeq Y \vee S^n$ implies $X \simeq Y$ for $(G,n)$-complexes $X$ and $Y$, and give lower bounds for the number of minimal homotopy types of $(G,n)$-complexes when this fails. The proof involves constructing projective $\mathbb{Z} G$ modules as lifts of locally free modules over orders in products of quaternion algebras, whose existence follows from the Eichler mass formula. In the case $n=2$, difficulties arise which lead to a new approach to finding a counterexample to Wall's D2 problem.Comment: 29 pages, comments welcom

Convergence Analysis of the Lowest Order Weakly Penalized Adaptive Discontinuous Galerkin Methods

In this article, we prove convergence of the weakly penalized adaptive discontinuous Galerkin methods. Unlike other works, we derive the contraction property for various discontinuous Galerkin methods only assuming the stabilizing parameters are large enough to stabilize the method. A central idea in the analysis is to construct an auxiliary solution from the discontinuous Galerkin solution by a simple post processing. Based on the auxiliary solution, we define the adaptive algorithm which guides to the convergence of adaptive discontinuous Galerkin methods

A three-dimensional application with the numerical grid generation code: EAGLE (utilizing an externally generated surface)

Program EAGLE (Eglin Arbitrary Geometry Implicit Euler) is a multiblock grid generation and steady-state flow solver system. This system combines a boundary conforming surface generation, a composite block structure grid generation scheme, and a multiblock implicit Euler flow solver algorithm. The three codes are intended to be used sequentially from the definition of the configuration under study to the flow solution about the configuration. EAGLE was specifically designed to aid in the analysis of both freestream and interference flow field configurations. These configurations can be comprised of single or multiple bodies ranging from simple axisymmetric airframes to complex aircraft shapes with external weapons. Each body can be arbitrarily shaped with or without multiple lifting surfaces. Program EAGLE is written to compile and execute efficiently on any CRAY machine with or without Solid State Disk (SSD) devices. Also, the code uses namelist inputs which are supported by all CRAY machines using the FORTRAN Compiler CF177. The use of namelist inputs makes it easier for the user to understand the inputs and to operate Program EAGLE. Recently, the Code was modified to operate on other computers, especially the Sun Spare4 Workstation. Several two-dimensional grid configurations were completely and successfully developed using EAGLE. Currently, EAGLE is being used for three-dimension grid applications